Summary

Polyharmonic B-splines are localized versions of radial basis functions. They can be designed to be nearly isotropic and Gaussian-like. They can also generate fractional wavelet bases on arbitrary lattices. The corresponding wavelets behave like multiscale fractional iterates of the Laplacian operator.

Main Contribution

Polyharmonic B-spline functions are non-separable, multidimensional basis functions that are localized versions of radial basis functions. We show that Rabut's elementary polyharmonic B-splines do not converge to a Gaussian as the order parameter increases, as opposed to their separable B-spline counterparts. Therefore, we introduce a more isotropic localization operator that guarantees this convergence, resulting into the isotropic polyharmonic B-splines.

Next, we use polyharmonic B-splines to build multidimensional wavelet bases. Therefore, we focus on the two-dimensional quincunx subsampling scheme. This configuration is of particular interest for image processing, because it yields a finer scale progression than the standard dyadic approach. However, up to now, the design of appropriate filters for the quincunx scheme has mainly been done using the McClellan transform. In our approach, we start from scaling functions that are the polyharmonic B-splines—thus their explicit form is known—to derive a family of polyharmonic spline wavelets that correspond to different flavors of the semi-orthogonal wavelet transform ( e . g ., orthonormal, B-spline, dual). The filters are automatically specified by the scaling relations satisfied by these functions. We prove that the isotropic polyharmonic B-spline wavelet converges to a combination of four Gabor atoms which are well-separated in the frequency domain. We also show that these wavelets are nearly isotropic and that they behave as an iterated Laplacian operator at low frequencies. We describe an efficient FFT-based implementation of the discrete wavelet transform based on polyharmonic B-splines.